Building Optimal Binary Search Trees from Sorted Values in O(N) Time

Abstract

First, we present a simple algorithm which, given a sorted sequence of node values, can build a binary search tree of minimum height in O(N) time. The algorithm works with sequences whose length is, a priori, unknown. Previous algorithms [1–3] required the number of elements to be known in advance. Although the produced trees are of minimum height, they are generally unbalanced. We then show how to convert them into optimal trees with a minimum internal path length in O(log N) time. The trees produced, both minimum height and optimal, have characteristic shapes which can easily be predicted from the binary representation of tree size.